inria-00195397https://inria.hal.science/inria-00195397https://inria.hal.science/inria-00195397v1/documenthttps://inria.hal.science/inria-00195397v1/file/discrete_non_det_report.pdfarxiv:0712.1519[ENS-LYON] École Normale Supérieure de Lyon[CNRS] CNRS - Centre national de la recherche scientifique[INRIA] INRIA - Institut National de Recherche en Informatique et en Automatique[UNIV-LYON1] Université Claude Bernard - Lyon I[PRUNEL] Ecole Normale Supérieure de Lyon[LIP] Laboratoire de l'Informatique du Parallélisme[LIP_PLUME] PLUME[LARA] LARA[INRIA-AUT] Inria autres[UDL] UDL[UNIV-LYON] Université de LyonDiscrete Non Neterminism and Nash Equilibria for Strategy-Based GamesLe Roux, StéphaneACM : G.: Mathematics of Computing/G.2: DISCRETE MATHEMATICSACM : F.: Theory of Computation/F.2: ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY/F.2.2: Nonnumerical Algorithms and Problems[INFO.INFO-GT] Computer Science [cs]/Computer Science and Game Theory [cs.GT]REPORTAbstract strategic gamesNash equilibriumdiscrete non-determinismdiscrete equilibriumconstructivefixed-pointmulti-strategic games.multi-strategic games
Several notions of game enjoy a Nash-like notion of equilibrium without guarantee of existence. There are different ways of weakening a definition of Nash-like equilibrium in order to guarantee the existence of a weakened equilibrium. Nash's approach to the problem for strategic games is probabilistic, \textit{i.e.} continuous, and static. CP and BR approaches for CP and BR games are discrete and dynamic. This paper proposes an approach that lies between those two different approaches: a discrete and static approach. multi strategic games are introduced as a formalism that is able to express both sequential and simultaneous decision-making, which promises a good modelling power. multi strategic games are a generalisation of strategic games and sequential graph games that still enjoys a Cartesian product structure, \textit{i.e.} where agent actually choose their strategies. A pre-fixed point result allows guaranteeing existence of discrete and non deterministic equilibria. On the one hand, these equilibria can be computed with polynomial (low) complexity. On the other hand, they are effective in terms of recommendation, as shown by a numerical example.
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